Doppler effect for sound waves

AS Level Physics · Waves. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

When a source of sound moves relative to a stationary observer, the observer hears a frequency different from the one the source gives out. This is the Doppler effect. The source itself does not change its frequency.

If the source moves towards the observer, each wavefront is sent out from a point closer to the observer than the last. The wavefronts are squeezed together, so the wavelength is shorter. The speed of sound in the air is unchanged, so the observed frequency is higher. If the source moves away, the wavelength is longer and the observed frequency is lower.

For a source passing close by at constant speed, the frequency is steady and high, then drops to a steady lower value.

Rules to remember

observed frequency = source frequency × speed of sound ÷ (speed of sound ± speed of source): f_o = f_sv/(v ± v_s)
Source moving towards the observer: use v − v_s, so f_o is higher than f_s
Source moving away from the observer: use v + v_s, so f_o is lower than f_s
The speed of the sound waves through the air does not change; the wavelength does
Ratio of approaching to receding frequency = (v + v_s)/(v − v_s)

Common mistake

Students choose the wrong sign. Check with common sense: a source coming towards you must give a higher frequency, so the bottom of the fraction must be smaller (v − v_s).

Worked example

A police car moves directly towards a stationary observer at 30 m s⁻¹. The observer hears a frequency of 640 Hz. The speed of sound is 330 m s⁻¹. What frequency does the siren give out?

Source approaching, so f_o = f_sv/(v − v_s)
Rearrange: f_s = f_o(v − v_s)/v = 640 × (330 − 30) ÷ 330
f_s = 640 × 300 ÷ 330 = 582 Hz

Answer: About 580 Hz (lower than the 640 Hz heard, as expected for an approaching source)

Practice questions

Try each one, then open the answer.

1. A sound source moves away from a stationary observer. Compared with the source frequency, the observed frequency is

A
higher, because the wavefronts are compressed
B
lower, because the wavefronts are stretched apart
C
the same
D
zero

Show answer

Answer: B. As the source recedes each successive wavefront is emitted from further away, increasing the wavelength reaching the observer and lowering the frequency.

2. A train sounds a horn of frequency 450 Hz as it moves directly away from a stationary observer at 25 m s⁻¹. The speed of sound is 340 m s⁻¹. What frequency does the observer hear?

A
417 Hz
B
419 Hz
C
483 Hz
D
486 Hz

Show answer

Answer: B. For a receding source f_o = f_sv/(v + v_s) = 450 × 340 ÷ 365 = 419 Hz. The frequency is lower.

3. An ambulance siren of frequency 600 Hz moves towards a stationary observer at 30 m/s. The speed of sound is 340 m/s. What frequency does the observer hear?

A
549 Hz
B
552 Hz
C
653 Hz
D
658 Hz

Show answer

Answer: D. f_o = f_s v/(v − v_s) = 600 × 340/(340 − 30) = 600 × 340/310 = 658 Hz.

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Doppler effect for sound waves

The idea

Rules to remember

Common mistake

Worked example

A police car moves directly towards a stationary observer at 30 m s−1. The observer hears a frequency of 640 Hz. The speed of sound is 330 m s−1. What frequency does the siren give out?

Practice questions

1. A sound source moves away from a stationary observer. Compared with the source frequency, the observed frequency is

2. A train sounds a horn of frequency 450 Hz as it moves directly away from a stationary observer at 25 m s−1. The speed of sound is 340 m s−1. What frequency does the observer hear?

3. An ambulance siren of frequency 600 Hz moves towards a stationary observer at 30 m/s. The speed of sound is 340 m/s. What frequency does the observer hear?

Read the full notes

Keep going

A police car moves directly towards a stationary observer at 30 m s⁻¹. The observer hears a frequency of 640 Hz. The speed of sound is 330 m s⁻¹. What frequency does the siren give out?

2. A train sounds a horn of frequency 450 Hz as it moves directly away from a stationary observer at 25 m s⁻¹. The speed of sound is 340 m s⁻¹. What frequency does the observer hear?